It was discovered by Étienne Halphen, who had become interested in the statistical modeling of natural events.

His practical experience in data analysis motivated him to pioneer a new system of distributions that provided sufficient flexibility to fit a large variety of data sets.

Halphen restricted his search to distributions whose parameters could be estimated using simple statistical approaches.

The harmonic law is a special case of the generalized inverse Gaussian distribution family when

One of Halphen's tasks, while working as statistician for Electricité de France, was the modeling of the monthly flow of water in hydroelectric stations.

Halphen realized that the Pearson system of probability distributions could not be solved; it was inadequate for his purpose despite its remarkable properties.

Therefore, Halphen's objective was to obtain a probability distribution with two parameters, subject to an exponential decay both for large and small flows.

[2] The harmonic law is the only one two-parameter family of distributions that is closed under change of scale and under reciprocals, such that the maximum likelihood estimator of the population mean is the sample mean (Gauss' principle).

[3] In 1946, Halphen realized that introducing an additional parameter, flexibility could be improved.

[4] where: Hence the mean and the succeeding three moments about it are Skewness is the third standardized moment around the mean divided by the 3/2 power of the standard deviation, we work with,[4] The coefficient of kurtosis is the fourth standardized moment divided by the square of the variance., for the harmonic distribution it is[4] The likelihood function is After that, the log-likelihood function is From the log-likelihood function, the likelihood equations are These equations admit only a numerical solution for a, but we have The mean and the variance for the harmonic distribution are,[3][4] Note that The method of moments consists in to solve the following equations: where

using The harmonic law is a sub-family of the generalized inverse Gaussian distribution.

This explains why the normal distribution can be used successfully for certain data sets of ratios.

This family has an interesting property, the Pitman estimator of the location parameter does not depend on the choice of the loss function.